For this one you have to escape by bribing the Warden.
The easiest way I found to do this was to use one of the prisons I'd made using architect and to make sure I made the wardens office and the areas leading up to it shared areas under security sectors.
I then just had to go about raising $1200 to bribe the warden (I'm not sure if this is a set amount or if it will vary from prison to prison).
I also did this on standard mode as there's a warning that comes up saying that most the achievements can only be obtained on it so I'm not sure if you can use an easier or custom setting to obtain it.
EDIT - As per the comments, the amount you need does vary and also note that the option to bribe the warden does not appear until you have sufficient money.